Modeling and Forecasting Realized Volatility with Multivariate Fractional Brownian Motion
提出用多元分数布朗运动（各分量有不同赫斯特指数）来建模和预测已实现波动率，推导了时间可逆条件下的最优预测公式，实证显示其预测误差低于一元模型和HAR模型。
A multivariate fractional Brownian motion (mfBm) with component-wise Hurst exponents is used to model and forecast realized volatility (RV). We investigate the interplay between correlation coefficients and Hurst exponents and propose a novel method to estimate model parameters, establishing its consistency and asymptotic normality. Additionally, we develop a time-reversibility test, which is typically not rejected by RV data. When the data generating process is a time-reversible mfBm, we derive optimal forecasting formulae and analyze their properties. A key insight is that an mfBm with different Hurst exponents and non-zero correlations can reduce forecasting errors compared to a one-dimensional model. Consistent with this theory, out-of-sample forecasts using the time-reversible mfBm show improvements over univariate fBm, particularly when the estimated Hurst exponents differ significantly. Empirical results demonstrate that mfBm outperforms HAR and its variants in terms of out-of-sample forecast.